Contents

model category

for ∞-groupoids

# Contents

## Idea

A category with weak equivalences is an ordinary category with a class of morphisms singled out – called ‘weak equivalences’ – that include the isomorphisms, but also typically other morphisms. Such a category can be thought of as a presentation of an (∞,1)-category that defines explicitly only the 1-morphisms (as opposed to n-morphisms for all $n$) and the information about which of these morphisms should become equivalences in the full (∞,1)-category.

The desired $(\infty,1)$-category in question can be constructed from such a “presentation” by “freely adjoining inverse equivalences” to the weak equivalences, in a suitable $(\infty,1)$-categorical sense. One way to make this precise is by the process of simplicial localization . A single $(\infty,1)$-category can admit many different such presentations. See the section Presentations of (∞,1)-categories below for more details.

## Definition

A category with weak equivalences is a category $C$ equipped with a subcategory (in the naïve sense) $W \subset C$

• which contains all isomorphisms of $C$;

• which satisfies two-out-of-three: for $f, g$ any two composable morphisms of $C$, if two of $\{f, g, g \circ f\}$ are in $W$, then so is the third.

## Examples and refinements

Often categories with weak equivalences are equipped with further extra structure that helps with computing the simplicial localization, the homotopy category and derived functors.

Other variants include

Three additional conditions which categories with weak equivalences often satisfy are:

In fact, these three conditions are closely related.

• Obviously, saturation implies closure under retracts and two-out-of-six, since the isomorphisms in any category satisfy both.

• In any model category, all three conditions hold automatically.

• If the weak equivalences admit a calculus of fractions, or a well-behaved class of cofibrations or fibrations, then the three conditions are equivalent. See two-out-of-six property for the proofs, which are from Categories and Sheaves (for the calculus of fractions) and Blumberg-Mandell (for the case of cofibrations, in the context of a Waldhausen category).

## Remarks

• If we denote by $Core(C)$ the core of $C$ – the maximal subgroupoid of $C$ – then we have a chain of inclusions $Core(C) \hookrightarrow W \hookrightarrow C$.

• Many categories with weak equivalences can be equipped with the further structure of a model category. On the other hand, some categories with weak equivalences can not be equipped with a useful structure of a model category. In particular, categories of diagrams in a model category do not always inherit a useful model structure (on the other hand often they do, see model structure on functors). Several concepts exist that weaken the axioms of a model category in order to still obtain useful results in such a case – for instance a category of fibrant objects.

• Although categories of weak equivalences do not usually have limits and colimits, they are often accessible, and can be presented as an injectivity class? or a cone-injectivity class?. This is used in Smith’s recognition theorem for combinatorial model categories and can be “algebraicized” as in Bourke17.

## Presentation of $(\infty,1)$-categories

A category $C$ with weak equivalences serves as a presentation of an (∞,1)-category $\mathbf{C}$ with the same objects and at least the 1-morphisms of $C$, and such that every weak equivalence in $C$ becomes a true equivalence (a homotopy equivalence) in $\mathbf{C}$.

The procedure (or one of its equivalent variants) that constructs the (∞,1)-category $\mathbf{C}$ from the category with weak equivalences $C$ is called Dwyer-Kan simplicial localization.

In fact, every (∞,1)-category may be presented this way (and indeed posets equipped with wide subcategories of morphisms called weak equivalences are sufficient). This is discussed at

Alternatively, we may further project to the 1-category in which all weak equivalences become true isomorphisms: this is the homotopy category of $C$ with respect to $W$. Equivalently this is the homotopy category of an (∞,1)-category of $\mathbf{C}$.

Note that the category with weak equivalences which presents a given $(\infty,1)$-category can not, in general, be taken to be the homotopy category of that $(\infty,1)$-category; more “flab” must be built into it.

It also cannot, in general, be the underlying 1-category of a simplicially enriched presentation of that $(\infty,1)$-category. For instance, every $\infty$-groupoid can be realized as a simplicially enriched groupoid, but the underlying 1-category of a simplicially enriched groupoid is a 1-groupoid, which cannot be localized any further to produce a non-1-truncated $\infty$-groupoid.

Algebraic model structures: Quillen model structures, mainly on locally presentable categories, and their constituent categories with weak equivalences and weak factorization systems, that can be equipped with further algebraic structure and “freely generated” by small data.

structuresmall-set-generatedsmall-category-generatedalgebraicized
weak factorization systemcombinatorial wfsaccessible wfsalgebraic wfs
model categorycombinatorial model categoryaccessible model categoryalgebraic model category
construction methodsmall object argumentsame as $\to$algebraic small object argument